Author's Comments:In this long paper, we apply the ideas from Part I
together with
the equivalence method to classify the parabolic PDE in
the plane that
admit conservation laws. We show, in particular, that a
parabolic PDE that
has more than 3 independent conservation laws is
linearizable by a (contact)
change of coordinates and exhibit equations (to our
knowledge, the first
known ones) of parabolic equations that have exactly 3
independent conservation
laws. In the final section of the paper, we prove a
classification theorem
for parabolic systems that admit non-trivial integrable
extensions (i.e.,
'coverings' in Vinogradov's terminology) and give
examples of systems that
admit non-trivial coverings but no conservation laws.

My former student, Jeanne Nielsen Clelland (now at
the University of
Colorado in Boulder), has now generalized many of
these results to the
case of two independent space variables and is
developing the theory very
nicely.